The discriminant of a quadratic equation $ax^2 + bx + c = 0$ is given by the expression:

$\Delta = b^2 - 4ac$

The value of the discriminant $\Delta$ determines the nature of the roots of the quadratic equation. Here's an easy way to remember the characteristics of the roots based on the value of the discriminant:

1. If $\Delta > 0$:

• Roots are real and distinct (two different real roots).
• The quadratic equation has two different solutions, and the graph of the parabola intersects the x-axis at two points.

2. If $\Delta = 0$:

• Roots are real and equal (one repeated real root).
• The quadratic equation has exactly one solution (a double root), and the graph of the parabola touches the x-axis at a single point (vertex lies on the x-axis).

3. If $\Delta < 0$:

• Roots are complex (no real roots).
• The quadratic equation has no real solutions, only two complex conjugate solutions. The graph of the parabola does not intersect the x-axis at all.

Summary:

• $\Delta > 0$: Two real and distinct roots.
• $\Delta = 0$: One real and repeated root.
• $\Delta < 0$: Two complex roots (no real roots).

This pattern is a quick way to determine the nature of the roots of any quadratic equation by simply calculating the discriminant.

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5 Related Questions:

1. How do you calculate the discriminant of a quadratic equation?
2. What is the significance of the discriminant in solving quadratic equations?
3. How does the discriminant relate to the graph of a quadratic function?
4. Can a quadratic equation have no real solutions? How does the discriminant help in identifying that?
5. How do the coefficients of a quadratic equation affect the discriminant?

Tip:

To solve quadratic equations more easily, always compute the discriminant first. It gives you a quick insight into what type of solutions to expect.